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Adaptive Control of Time-delayed Systems with Application for Control of Glucose Concentration in Type 1 Diabetic Patients
11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France
ThS5T1.3
Adaptive Control of Time-delayed Systems with Application for Control of Glucose Concentration in Type 1 Diabetic Patients ⋆ Marián Tárník ∗ ∗
Ján Murgaš ∗ Eva Miklovičová ∗ Ľudovít Farkas ∗
Institute of Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Ilkovičova 3, 812 19 Bratislava, Slovakia. (e-mail: {marian.tarnik, jan.murgas, eva.miklovicova, ludovit.farkas}@stuba.sk)
Abstract: An adaptive controller for glucose control in Type 1 Diabetes Mellitus (T1DM) subject is presented in this paper. The proposed control model of T1DM subject involves a known input time-delay, due to the modeling of a subcutaneous tissues, and a disturbance submodel, where a meal ingestion acts as a measured disturbance. A main MRAC based part of controller for time-delayed systems is supplemented with a heuristic based adaptive disturbance rejection. The controller is verified by means of numerical simulations using an own implementation of T1DM simulator reported in literature. Keywords: adaptive control, input time-delay, type one diabetes mellitus, glucose control 1. INTRODUCTION Closed-loop control systems for maintaining a normoglicemia in diabetic patients have been subject of research since 1970s. However, the first devices had used intravenous blood glucose sampling and intravenous insulin and glucose delivery. In recent years a minimally invasive close-loop control system is under extensive research Magni et al. (2009). Such a system uses a subcutaneous continuous glucose monitoring and subcutaneous insulin delivery, which makes the control problem more challenging. In this control problem, the main challenges are the time delays, meal disturbances and the uncertain or unknown plant dynamics. Adaptive control is an effective approach for controlling uncertain systems with time delays. Numerous algorithms have been developed for the state delayed systems. In this paper we are focused on the plants with the input time delays as discussed below. In Yildiz et al. (2010); Niculescu and Annaswamy (2003) a model reference adaptive control (MRAC) approach for SISO input delayed plants has been used. More precisely, ⋆ This work has been supported by Slovak scientific grant agency through grant VEGA-1/2256/12. This paper is the result of implementation of the project: “Centre of competence for intelligent technologies for computerization and informatization of systems and services” (ITMS: 26240220072) supported by the Research & Development Operational Programme funded by the ERDF Podporujeme výskumné aktivity na Slovensku. Projekt je spolufinancovaný zo zdrojov EÚ.
978-3-902823-37-3/2013 © IFAC
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an output-feedback adaptive control algorithm for timedelayed systems with relative degree n⋆ ≤ 2 has been developed. In this algorithm a perfect model following in the sense of the classical MRAC is achieved. This is possible due to a Smith-predictor-like solution, where the plant output is adaptively predicted and this signal is utilized as the feedback signal in the control law. For this purpose the plant input delay is assumed to be known, while the remaining plant parameters are unknown. The plant output is predicted using the so-called distributeddelay block, which is a finite-time integral of the delayed control signal. To implement the control law, this integral is usually discretized, which leads to many difficulties, see Tarnik (2012), and only the model following (not perfect) can be practically achieved. Earlier results in this class of adaptive algorithms have been reported in Ortega and Lozano (1988). In Mirkin and Gutman (2011), the adaptive following algorithm for plants with input and state delays is proposed. This algorithm utilizes a classical Smith predictor, but based on the reference model transfer function rather than on the plant transfer function. The controlled system is therefore assumed to be stable (or stabilizable with socalled memoryless state feedback, see Mirkin et al. (2009)) and the input delay is assumed to be known. This allows to pull the input delay out of the design procedure. As a consequence, there is no need to use the distributeddelay blocks to establish a stability of the overall closedloop system. The price for this advantage is that only the model following can be theoretically achieved. Since the mentioned algorithm considers also the state delays, the state feedback is employed. However, we are interested in the output feedback control algorithm, which can be consequently applied to the control of glucose concentration. 10.3182/20130703-3-FR-4038.00033
11th IFAC ALCOSP July 3-5, 2013. Caen, France
rate of meal ingestion
2. CONTROLLED SYSTEM MODEL The controlled system with two inputs and one output is considered, see Fig. 1. The output is the subcutaneous glucose concentration ∆GM (t) [mmol/l]. The first input is the control input, particularly the insulin infusion rate (as a function of time t) ∆v(t) [pmol/kg/min]. The second input is the disturbance, in other words it is a meal ingestion rate ∆d(t) [mg/min]. The meal announcement information availability is considered as is common even in the conventional diabetes therapy Magni et al. (2009). All of the mentioned quantities are the deviations from the operating point given by vb , db , with corresponding GM b , where the subscript b denotes a basal state. Therefore ∆v(t) = v(t) − vb , ∆d(t) = d(t) − db and ∆GM (t) = Gm (t) − GM b . The basal insulin infusion rate vb maintains the glycemia at the basal value GM b , which can be for instance the value in the middle of a normal glycemia range 3.8 – 10 mmol/l. For an identification purpose, the discrete-time linear model is considered in the form Ad (z −1 )∆GM (k) = B1 (z −1 )∆v(k) + B2 (z −1 )∆d(k) (1) To determine the orders of polynomials Ad (z −1 ), B1 (z −1 ) and B2 (z −1 ), the simulation experiments have been performed (not reported in this paper). The T1DM subject simulator has been used to generate the identification data. A sufficiently accurate results have been obtained for Ad (z −1 ) = 1+ad1 z −1 +ad2 z −2 +ad3 z −3 , B1 (z −1 ) = b11 z −5 and B2 (z −1 ) = b21 z −1 + b22 z −2 + b23 z −3 . The model (1) can be written in the form b11 z −1 z −4 ∆v(k) + κ(k) (2) ∆GM (k) = −1 Ad (z ) where b11 z −1 κ(k) = B2 (z −1 )∆d(k). The aim is to obtain a continuous-time model. Moreover, the interpretation of the model parameters should be similar as in the case of a glucose kinetics minimal model, see Hovorka et al. (2002). The linearization of minimal model leads to the second order system with no zeros, for example see Miklovičová and Tárník (2012). Therefore, the discrete transfer function (2) has been converted to the continuous-time domain. Further, a model order reduction procedure has been applied to obtain the desired model form. Particularly, the discrete transfer function has been converted as follows: b11 z −1 b0 ⇒ 2 −1 Ad (z ) s + a1 s + a0 The first component of the overall input signal in (2) is simply the time-delayed insulin input, z −4 ∆v(k) ≈ e−τ s ∆u(t), where τ can be seen as a time-delay caused by a subcutaneous tissues. The second input component in (2) can be converted as follows: As mentioned above, we have b11 z −1 κ(k) = b21 z −1 + b22 z −2 + b23 z −3 ∆d(k). Therefore, we can write b11 κ(k) = b21 ∆d(k) + b22 ∆d(k − 1) + b23 ∆d(k − 2), which implies κ(t) = Ψ⋆ T w1 (t)
(3)
2i for i = 1, 2, 3 and where Ψ⋆ T = [ψ1⋆ ψ2⋆ ψ3⋆ ], ψi⋆ = bb11 T w1 (t) = [∆d(t) ∆d(t − τ1 ) ∆d(t − τ2 )], τi = i × Ts for i = 1, 2 and Ts is known sampling period used in the
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Fig. 1. Controlled system original discrete model (1). Notice, that τ = 4 × Ts is also known. Further, the disturbance signal availability is assumed in the form of the meal announcement information. Therefore, the future evolution of disturbance is assumed to be known. This allows to define wT (t) = [∆d(t + τ ) ∆d(t − τ1 + τ ) ∆d(t − τ2 + τ )]. Finally the continuous-time transfer function of controlled system has the form b0 ∆GM (s) = 2 (4) e−τ s ∆v(t) + Ψ⋆ T w(t) s + a1 s + a0 The simulations have shown that suitability of model (4) is similar to the original discrete model (1). 3. ADAPTIVE CONTROLLER DESIGN The proposed controller consists of two parts. First a classical model reference adaptive control (MRAC) based controller and second a disturbance rejection controller. The controller design is briefly presented in this section. 3.1 MRAC based controller To design the first part of the controller no disturbance is considered in the controlled system (4), i.e. Ψ⋆ T w(t) = 0. Therefore, the controlled system is in the form y(s) = W (s)e−τ s u(s) (5) where y(s) and u(s) is the output and input respectively. It is assumed that the relative degree of the transfer function W (s) is n⋆ = n − m = 2, and the time-delay τ is known.
System (5) can be written in the state-space form x(t) ˙ = Ax(t) + bu(t − τ )
(6a)
T
y(t) = c x(t) (6b) n×n n n where the matrices A ∈ R , b ∈ R and c ∈ R are unknown.
The auxiliary filters are introduced in the form ν˙ 1 (t) = Λν1 (t) + qu(t − τ ) (7a) ν˙ 2 (t) = Λν2 (t) + qy(t) (7b) where ν1 (t), ν2 (t) ∈ Rn−1 , q ∈ Rn−1 , q T = [0 · · · 0 1] and Λ ∈ Rn−1×n−1 is an arbitrary stable matrix. Equations (6) and (7) can be written in the compact form ˙ X(t) = Ao X(t) + Bc u(t − τ ) y(t) =
CcT X(t)
(8a) (8b)
11th IFAC ALCOSP July 3-5, 2013. Caen, France
where X T (t) = X T (t) ν1T (t) ν2T (t) and matrices Ao , Bc and Cc have an appropriate form. It is further assumed that there exist a constant vector Θ⋆c ∈ R2n−1 and a constant scalar Θ⋆4 ∈ R such that the following conditions hold Ac = Ao + Bc Θ⋆c T D
and
B c = Bc Θ⋆4
(9)
n−1×n−1
T
where D = diag(c , I, I), I ∈ R is an identity matrix and Ac ∈ R3n−2×3n−2 , B c ∈ R3n−2 are the matrices of nonminimal realization of the reference model in the form X˙ m (t) = Am Xm (t) + B c r(t) (10a) ym (t) = CcT Xm (t) (10b) where r(t) is the reference signal and ym (t) is the reference model output. The reference model transfer function ym (s) ⋆ r(s) = Wm (s) has the relative degree nm = 2. The equation (8a) represents the controlled system. The term Bc Θ⋆c T DX(t) and the term Bc Θ⋆4 r(t) are added and subtracted to (8a), then ˙ X(t) = Ac X(t)+B c r(t)−Bc Θ⋆ T ω(t) +Bc u(t−τ ) (11) where Θ⋆ T = Θ⋆c T Θ⋆4 and ω T (t) = X T (t)DT r(t) has been introduced. Further the term Bc u(t) and the term B c ρ(t)uτ (t), where uτ (t) = u(t) − u(t − τ ) and ρ(t) ∈ R is a control algorithm parameter to be adapted, are added and subtracted to (11), which implies ˙ X(t) = Ac X(t) + B c r(t) + Bc u(t) − Θ⋆ T ω(t) (12) + B c ρ(t)uτ (t) − Θ⋆4 −1 uτ (t) − B c ρ(t)uτ (t) The proposed control law has the form u(t) = ΘT (t)ω(t) (13) 2n−1 where Θ(t) ∈ R is the vector of adapted parameters. Substituting (13) to (12) leads to ˙ X(t) = Ac X(t) + B c r(t) − B c ρ(t)uτ (t) ⋆T ⋆ −1 B (t) − Θ ρ(t) − Θ + B c ΘT ω(t) + uτ (t) c τ τ 4
(14)
Θ⋆4 −1 Θ(t)
Θ⋆τ
where Θτ (t) = and is defined analogically. Further, the adapted parameter errors are defined as ˜ τ (t) = Θτ (t) + Θ⋆ and ρ˜(t) = ρ(t) − Θ⋆ −1 . Moreover Θ τ 4 T ˜ τ (t) ρ˜(t) and ωτT (t) = ω T (t) uτ (t) are θτT (t) = Θ introduced. Then the equation (14) has the form ˙ X(t) = Ac X(t) + B c r(t) − B c ρ(t)uτ (t) + B c θτT (t)ωτ (t) (15) Since the controlled system relative degree equals two, the augmented reference model model is considered. This is well-known approach in the classical MRAC design. Equation (10a) is considered in the form X˙ m (t) = Am Xm (t) (16) T + B c r(t) + θτ (t) − L(s)θτ (t)L−1 (s) ωτ (t) where L(s) = (s + ς), ς ∈ R, ς > 0 is specified below.
Subtracting (16) from (15), the error equation is obtained in the form 454
e(t) ˙ = Ac e(t) + B c ρ(t)uτ (t) + B c θτT (t)ωτ (t) T − B c θτ (t) − L(s)θτ (t)L−1 (s) ωτ (t)
(17)
where e(t) = X(t)−Xm (t). Introducing the transformation ea (t) = e(t) + Xa (t) allows to split (17) into two equations e˙ a (t) = Ac ea (t) + B c θτT (t)ωτ (t) − B c θτ (t) − L(s)θτ (t)L−1 (s)
T
ωτ (t)
(18)
(19) X˙ a (t) = Ac Xa (t) + B c ρ(t)uτ (t) where a Smith-predictor-like filter has been introduced. The output equation of the filter has the form ya (t) = CcT Xa (t). Equation (18) can be rearranged to the form e˙ a (t) = Ac ea (t) + B c L(s)θτ T (t)L−1 (s) ωτ (t) (20a) ea1 (t) = CcT ea (t) (20b) where the output equation has been added. The system (20) is described by the transfer function in the form ea1 (s) = Wm (s) L(s)θτ T (s)L−1 (s) ωτ (s) (21) At this point a standard error equation suitable for a standard Lyapunov based output-feedback MRAC design has been obtained. Equation (21) can be written in the form ea1 (s) = Wm (s)L(s) θτ T (s)ωf (s) (22) where ωf (s) = L−1 (s)ωτ (s) and ς is chosen so that the Wm (s)L(s) is a Strictly Positive Real (SPR) transfer function. In view of (22) the control objective can be considered as follows: designing an adaptation law so that the augmented error ea1 (t) tends to zero as t → ∞ implies that the standard error e1 (t) = y(t) − ym (t) = CcT e(t) is bounded. Recall that the controlled system is assumed to be stable. This is referred as an adaptive model following (not a perfect model following). Using a well-understood adaptive control methods (Lyapunov based) it can be shown that the adaptation law in the form θ˙τ (t) = −Γea1 (t)ωf (t) (23) ensures that the control objective is satisfied and all of the closed-loop signals are bounded. In (23) Γ = diag (Γτ , γ), Γτ > 0 is an arbitrary diagonal matrix of appropriate ˙ τ (t) = dimension and γ > 0 is a scalar. It follows that Θ −Γτ ea1 (t)ωf f (t) and ρ(t) ˙ = −γea1 (t)uf f (t) (24) where uf f (s) = L−1 (s)uτ (s) and ωf f (s) = L−1 (s)ω(s). As defined above Θ(t) = Θ⋆4 Θτ (t), which implies ˙ Θ(t) = −sgn(Θ⋆4 )Γτ ea1 (t)ωf f (t) (25)
Therefore, a sign of Θ⋆4 has to be known, in other words the sign of controlled system high-frequency gain has to be known.
It can be shown that the signal ea1 (t) can be implemented in the form ea1 (t) = y(t) − ym (t) + ya (t) − [Wm (s)L(s)] sgn(Θ⋆4 ) L−1 (s) u(t) (26) −sgn(Θ⋆4 )ΘT (t)ωf f (t) + L−1 (s) ρ(t)uτ (t) −ρ(t)uf f (t))
11th IFAC ALCOSP July 3-5, 2013. Caen, France
3.2 Modification for Robust Adaptive Control The disturbance has not been considered in the previous section. However, since the disturbance is not rejected completely in the considered controlled system, the design of adaptive controller has to be modified to handle the disturbance influence. A switching σ-modification, which is well known approach in robust adaptive control, is considered. Therefore, the adaptation laws (24) and (25) are modified as follows. ρ(t) ˙ = σρ ρ(t) − γea1 (t)uf f (t) (27) ⋆ ⋆ ˙ Θ(t) = sgn(Θ4 )σΘ Θ(t) − sgn(Θ4 )Γτ ea1 (t)ωf f (t) (28)
where
σρ = and
0 if |ρ(t)| ≤ ρmax σρ0 otherwise
(29)
0 if |Θ(t)| ≤ Θmax (30) σΘ0 otherwise where ρmax , σρ0 , Θmax and σΘ0 are the design constants. σΘ =
3.3 A Heuristic Based Adaptive Disturbance Rejection Recall the definition ea (t) = e(t) + Xa (t). It follows that ea1 (t) = y(t) − ym (t) + ya (t), and finally ym (t) = y(t) + ya (t) − ea1 (t) (31) However, if the disturbance is present, i.e. Ψ⋆ T w(t) 6= 0 in (4), the output signal y(t) in (31) is impaired. Consequently, the signal ym (t) obtained using (31) is also impaired, which is caused only by the disturbance.
An ideal (non-impaired) case is represented by the reference model transfer function ⋆ ym (s) = Wm (s)r(s) (32) Therefore, a heuristic error can be defined in the form ⋆ ed (t) = ym (t) − ym (t) (33) It is clear, that if ed (t) = 0 the disturbance is not present or it is rejected. The adaptive disturbance rejection control law is proposed in the form ud (t) = ΨT (t)w(t) (34) 3 where Ψ(t) ∈ R are the parameters to be adapted. A control action ud (t) is subtracted from u(t), in other words, in (4) it can be written ∆v(t) = u(t) − ΨT (t)w(t). In view of (33) the adaptation law is proposed in the form ˙ Ψ(t) = σΨ Ψ(t) − γd ed (t)w(t) (35)
where γd ∈ R, γd > 0 and 0 if |Ψ(t)| ≤ Ψmax σΨ = (36) σΨ0 otherwise where Ψmax and σΨ0 are the design constants. The adaptation law (35) can be considered as a gradient based and modified via switching σ-modification. 4. MAIN RESULTS The proposed adaptive controller has been tuned and verified by means of the simulation experiments. The T1DM subject computer simulator has been used as the controlled system.
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The simulator is an own implementation of A Meal Glucose-Insulin Model by Chiara Dalla Man and coworkers, see Man et al. (2007); De Nicolao et al. (2011). As has been mentioned above the same simulator has been used for the identification of the controlled system model. Sampling period Ts = 15 min has been used in (1). Therefore the time-delays τ = 60 min, τ1 = 15 min and τ2 = 30 min are assumed to be known to the controller. The main objective in diabetes control is to keep the blood glucose concentration as tightly as possible within a normal glycemic range (3.8 – 10 mmol/l). The reference model used in the adaptive controller is chosen in the form 1.6 × 10−5 ym (s) = 2 r(s) (37) s + 0.008s + 1.6 × 10−5 i.e. two poles at −0.004 min−1 . As the reference signal r(t) a periodic square signal is used, with an amplitude 0.5 mmol/l (around the operating point GM b ) and with the period one day. Simulations are started in the steady state with the basal subcutaneous glucose concentration GM b . The tuning parameters of the adaptive controller such as the adaptation gains or the auxiliary filters are the same in all simulation experiments. The tuning parameters and the reference model have been chosen experimentally so that the same set can be used for each task considered in all simulations. A time period of three weeks is simulated. 4.1 Experiment No. 1 A control performance of the proposed controller with no disturbance present has been examined in the first simulation experiment. Therefore the heuristic based adaptive disturbance rejection has been turned off. Since the disturbance has not been present the switching σ-modification could be unused. In this experiment only the MRAC based part of controller has been used. The initial values of corresponding adapted parameters have been chosen experimentally and the same initial values have been used in all experiments. Results are shown in Fig. 2. As the figure clearly shows, the transient process of adaptation is over in three weeks. The model following (recall that the perfect model following has not been the control objective) has been achieved while the influence of the time-delay has been handled by controller. These results have verified the design of the MRAC based controller. However, a considered situation is not common from the diabetes control point of view. 4.2 Experiment No. 2 In the second experiment the disturbance has been considered in the form of meal ingestion. Each day the same meal protocol has been used as follows: a typical day life of the T1DM subject receiving a mixed meal, with 30 g of glucose over 15 minutes ingested at 8 a.m. (breakfast), 50 g of glucose over 15 minutes at noon (lunch) and 50 g of glucose over 15 minutes at 8 p.m. (dinner). This situation is more common in the diabetes control than the first experiment. However, the same controller setup as in the first experiment has been used, i.e. the heuristic based adaptive
11th IFAC ALCOSP July 3-5, 2013. Caen, France
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Fig. 4. Results of the simulation experiment No. 3 disturbance rejection has been turned off. Since the obvious presence of the disturbance, the use of switching σ-modification has been justified in this case. This has also guarantee the robust stability of the overall closedloop system. The control performance, however, has to be verified. The results are shown in Fig. 3. It can be seen, that the main control objective has not been satisfied, i.e. the glucose concentration in the normal glycemic range. Simulations for longer time interval than three weeks has shown, that the best performance, in the considered 456
situation, has been achieved in th 21st day and it has remained unchanged in next weeks. Moreover, during the adaptation the hypoglycemia has occurred in the first nine days, as emphasized in Fig. 3a. Nevertheless, it can be concluded that the performance after the tenth day is comparable to the diabetes control algorithms without a meal announcement considered (for example the classical PID controllers or the generalized predictive control algorithms reported in literature, see Miklovičová and Tárník (2012)).
11th IFAC ALCOSP July 3-5, 2013. Caen, France
4.3 Experiment No. 3 The same meal protocol as in the second experiment has been considered in the third experiment. The adaptive disturbance rejection, however, has been used in this experiment. Results in Fig. 4 have shown that after the adaptation transient, which has last only one day, the main control objective has been achieved. It can be concluded, that the heuristic adaptive disturbance rejection has a significant influence on the overall control performance. 5. CONCLUSION AND FUTURE WORK The presented adaptive controller design is based on the controlled system model inferred from the glucose kinetics minimal model, where in addition the subcutaneous tissues are modeled using the known input time-delay. The main results have shown that for the exact knowledge of this time-delay a satisfactory glucose control is obtained and the control objective is achieved. However, the influence of uncertainty in this time-delay has to be examined in future research. Our recent results have shown that for the stability the considered robust modification is adequate. Moreover, the control performance deterioration has shown to be acceptable from the diabetes control point of view. The same is true for the developed disturbance submodel, since there are also the time-delays, which are assumed to be known. However, in this case the uncertainty in the announced meal time has to be also considered in the future research. Nevertheless, the well-know importance of the disturbance rejection in the algorithms for diabetes control has been also shown in the main results. The benefits of the adaptive disturbance rejection in this case are obvious. REFERENCES De Nicolao, G., Magni, L., Dalla Man, C. and Cobelli, C. (2011), Modeling and control of diabetes: Towards the artificial pancreas, in ‘Proceedings of the 18th IFAC World Congress’. Hovorka, R., Shojaee-Moradie, F., Carroll, P. V., Chassin, L. J., Gowrie, I. J., Jackson, N. C., Tudor, R. S., Umpleby, A. M. and Jones, R. H. (2002), ‘Partitioning glucose distribution/transport, disposal, and endogenous production during ivgtt’, American Journal of Physiology - Endocrinology and Metabolism (282), E992–E1007. Magni, L., Raimondo, D., Man, C. D., Nicolao, G. D., Kovatchev, B. and Cobelli, C. (2009), ‘Model predictive control of glucose concentration in type i diabetic patients: An in silico trial’, Biomedical Signal Processing and Control 4(4), 338 – 346. Special Issue on Biomedical Systems, Signals and Control Extended Selected papers from the IFAC World Congress, Seoul, July 2008. Man, C., Raimondo, D., Rizza, R. and Cobelli, C. (2007), ‘Gim, simulation software of meal glucose–insulin model’, Journal of Diabetes Science and Technology 1(3). 457
Miklovičová, E. and Tárník, M. (2012), GPC for diabetes control without meal annoucement — control loop design and control performance study, in O. Arslan and S. Oprisan, eds, ‘Recent Advances in Mechanical Engineering and Automatic Control, Proceedings of the 3rd European Conference of Control (ECC ’12), Paris, France December 2-4, 2012’, WSEAS Press, pp. 58 – 63. Mirkin, B. and Gutman, P.-O. (2011), Adaptive following of perturbed plants with input and state delays, in ‘2011 9th IEEE International Conference on Control and Automation (ICCA)’, pp. 865 –870. Mirkin, B., Mirkin, E. L. and Gutman, P.-O. (2009), ‘State-feedback adaptive tracking of linear systems with input and state delays’, International Journal of Adaptive Control and Signal Processing 23(6), 567–580. Niculescu, S. and Annaswamy, A. M. (2003), ‘An adaptive smith-controller for time-delay systems with relative degree n⋆ ≤ 2’, Systems and Control Letters 49(5), 347 – 358. Ortega, R. and Lozano, R. (1988), ‘Globally stable adaptive controller for systems with delay’, International Journal of Control 47, 17–23. Tarnik, M. (2012), On adaptive posi-cast controller: Pmsm speed controller design, in ‘10th International Conference CONTROL OF POWER SYSTEMS 2012’. Yildiz, Y., Annaswamy, A., Kolmanovsky, I. V. and Yanakiev, D. (2010), ‘Adaptive posicast controller for time-delay systems with relative degree n∗ ≤ 2’, Automatica 46(2), 279 – 289.
ThS5T1.3
Adaptive Control of Time-delayed Systems with Application for Control of Glucose Concentration in Type 1 Diabetic Patients ⋆ Marián Tárník ∗ ∗
Ján Murgaš ∗ Eva Miklovičová ∗ Ľudovít Farkas ∗
Institute of Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Ilkovičova 3, 812 19 Bratislava, Slovakia. (e-mail: {marian.tarnik, jan.murgas, eva.miklovicova, ludovit.farkas}@stuba.sk)
Abstract: An adaptive controller for glucose control in Type 1 Diabetes Mellitus (T1DM) subject is presented in this paper. The proposed control model of T1DM subject involves a known input time-delay, due to the modeling of a subcutaneous tissues, and a disturbance submodel, where a meal ingestion acts as a measured disturbance. A main MRAC based part of controller for time-delayed systems is supplemented with a heuristic based adaptive disturbance rejection. The controller is verified by means of numerical simulations using an own implementation of T1DM simulator reported in literature. Keywords: adaptive control, input time-delay, type one diabetes mellitus, glucose control 1. INTRODUCTION Closed-loop control systems for maintaining a normoglicemia in diabetic patients have been subject of research since 1970s. However, the first devices had used intravenous blood glucose sampling and intravenous insulin and glucose delivery. In recent years a minimally invasive close-loop control system is under extensive research Magni et al. (2009). Such a system uses a subcutaneous continuous glucose monitoring and subcutaneous insulin delivery, which makes the control problem more challenging. In this control problem, the main challenges are the time delays, meal disturbances and the uncertain or unknown plant dynamics. Adaptive control is an effective approach for controlling uncertain systems with time delays. Numerous algorithms have been developed for the state delayed systems. In this paper we are focused on the plants with the input time delays as discussed below. In Yildiz et al. (2010); Niculescu and Annaswamy (2003) a model reference adaptive control (MRAC) approach for SISO input delayed plants has been used. More precisely, ⋆ This work has been supported by Slovak scientific grant agency through grant VEGA-1/2256/12. This paper is the result of implementation of the project: “Centre of competence for intelligent technologies for computerization and informatization of systems and services” (ITMS: 26240220072) supported by the Research & Development Operational Programme funded by the ERDF Podporujeme výskumné aktivity na Slovensku. Projekt je spolufinancovaný zo zdrojov EÚ.
978-3-902823-37-3/2013 © IFAC
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an output-feedback adaptive control algorithm for timedelayed systems with relative degree n⋆ ≤ 2 has been developed. In this algorithm a perfect model following in the sense of the classical MRAC is achieved. This is possible due to a Smith-predictor-like solution, where the plant output is adaptively predicted and this signal is utilized as the feedback signal in the control law. For this purpose the plant input delay is assumed to be known, while the remaining plant parameters are unknown. The plant output is predicted using the so-called distributeddelay block, which is a finite-time integral of the delayed control signal. To implement the control law, this integral is usually discretized, which leads to many difficulties, see Tarnik (2012), and only the model following (not perfect) can be practically achieved. Earlier results in this class of adaptive algorithms have been reported in Ortega and Lozano (1988). In Mirkin and Gutman (2011), the adaptive following algorithm for plants with input and state delays is proposed. This algorithm utilizes a classical Smith predictor, but based on the reference model transfer function rather than on the plant transfer function. The controlled system is therefore assumed to be stable (or stabilizable with socalled memoryless state feedback, see Mirkin et al. (2009)) and the input delay is assumed to be known. This allows to pull the input delay out of the design procedure. As a consequence, there is no need to use the distributeddelay blocks to establish a stability of the overall closedloop system. The price for this advantage is that only the model following can be theoretically achieved. Since the mentioned algorithm considers also the state delays, the state feedback is employed. However, we are interested in the output feedback control algorithm, which can be consequently applied to the control of glucose concentration. 10.3182/20130703-3-FR-4038.00033
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rate of meal ingestion
2. CONTROLLED SYSTEM MODEL The controlled system with two inputs and one output is considered, see Fig. 1. The output is the subcutaneous glucose concentration ∆GM (t) [mmol/l]. The first input is the control input, particularly the insulin infusion rate (as a function of time t) ∆v(t) [pmol/kg/min]. The second input is the disturbance, in other words it is a meal ingestion rate ∆d(t) [mg/min]. The meal announcement information availability is considered as is common even in the conventional diabetes therapy Magni et al. (2009). All of the mentioned quantities are the deviations from the operating point given by vb , db , with corresponding GM b , where the subscript b denotes a basal state. Therefore ∆v(t) = v(t) − vb , ∆d(t) = d(t) − db and ∆GM (t) = Gm (t) − GM b . The basal insulin infusion rate vb maintains the glycemia at the basal value GM b , which can be for instance the value in the middle of a normal glycemia range 3.8 – 10 mmol/l. For an identification purpose, the discrete-time linear model is considered in the form Ad (z −1 )∆GM (k) = B1 (z −1 )∆v(k) + B2 (z −1 )∆d(k) (1) To determine the orders of polynomials Ad (z −1 ), B1 (z −1 ) and B2 (z −1 ), the simulation experiments have been performed (not reported in this paper). The T1DM subject simulator has been used to generate the identification data. A sufficiently accurate results have been obtained for Ad (z −1 ) = 1+ad1 z −1 +ad2 z −2 +ad3 z −3 , B1 (z −1 ) = b11 z −5 and B2 (z −1 ) = b21 z −1 + b22 z −2 + b23 z −3 . The model (1) can be written in the form b11 z −1 z −4 ∆v(k) + κ(k) (2) ∆GM (k) = −1 Ad (z ) where b11 z −1 κ(k) = B2 (z −1 )∆d(k). The aim is to obtain a continuous-time model. Moreover, the interpretation of the model parameters should be similar as in the case of a glucose kinetics minimal model, see Hovorka et al. (2002). The linearization of minimal model leads to the second order system with no zeros, for example see Miklovičová and Tárník (2012). Therefore, the discrete transfer function (2) has been converted to the continuous-time domain. Further, a model order reduction procedure has been applied to obtain the desired model form. Particularly, the discrete transfer function has been converted as follows: b11 z −1 b0 ⇒ 2 −1 Ad (z ) s + a1 s + a0 The first component of the overall input signal in (2) is simply the time-delayed insulin input, z −4 ∆v(k) ≈ e−τ s ∆u(t), where τ can be seen as a time-delay caused by a subcutaneous tissues. The second input component in (2) can be converted as follows: As mentioned above, we have b11 z −1 κ(k) = b21 z −1 + b22 z −2 + b23 z −3 ∆d(k). Therefore, we can write b11 κ(k) = b21 ∆d(k) + b22 ∆d(k − 1) + b23 ∆d(k − 2), which implies κ(t) = Ψ⋆ T w1 (t)
(3)
2i for i = 1, 2, 3 and where Ψ⋆ T = [ψ1⋆ ψ2⋆ ψ3⋆ ], ψi⋆ = bb11 T w1 (t) = [∆d(t) ∆d(t − τ1 ) ∆d(t − τ2 )], τi = i × Ts for i = 1, 2 and Ts is known sampling period used in the
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Fig. 1. Controlled system original discrete model (1). Notice, that τ = 4 × Ts is also known. Further, the disturbance signal availability is assumed in the form of the meal announcement information. Therefore, the future evolution of disturbance is assumed to be known. This allows to define wT (t) = [∆d(t + τ ) ∆d(t − τ1 + τ ) ∆d(t − τ2 + τ )]. Finally the continuous-time transfer function of controlled system has the form b0 ∆GM (s) = 2 (4) e−τ s ∆v(t) + Ψ⋆ T w(t) s + a1 s + a0 The simulations have shown that suitability of model (4) is similar to the original discrete model (1). 3. ADAPTIVE CONTROLLER DESIGN The proposed controller consists of two parts. First a classical model reference adaptive control (MRAC) based controller and second a disturbance rejection controller. The controller design is briefly presented in this section. 3.1 MRAC based controller To design the first part of the controller no disturbance is considered in the controlled system (4), i.e. Ψ⋆ T w(t) = 0. Therefore, the controlled system is in the form y(s) = W (s)e−τ s u(s) (5) where y(s) and u(s) is the output and input respectively. It is assumed that the relative degree of the transfer function W (s) is n⋆ = n − m = 2, and the time-delay τ is known.
System (5) can be written in the state-space form x(t) ˙ = Ax(t) + bu(t − τ )
(6a)
T
y(t) = c x(t) (6b) n×n n n where the matrices A ∈ R , b ∈ R and c ∈ R are unknown.
The auxiliary filters are introduced in the form ν˙ 1 (t) = Λν1 (t) + qu(t − τ ) (7a) ν˙ 2 (t) = Λν2 (t) + qy(t) (7b) where ν1 (t), ν2 (t) ∈ Rn−1 , q ∈ Rn−1 , q T = [0 · · · 0 1] and Λ ∈ Rn−1×n−1 is an arbitrary stable matrix. Equations (6) and (7) can be written in the compact form ˙ X(t) = Ao X(t) + Bc u(t − τ ) y(t) =
CcT X(t)
(8a) (8b)
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where X T (t) = X T (t) ν1T (t) ν2T (t) and matrices Ao , Bc and Cc have an appropriate form. It is further assumed that there exist a constant vector Θ⋆c ∈ R2n−1 and a constant scalar Θ⋆4 ∈ R such that the following conditions hold Ac = Ao + Bc Θ⋆c T D
and
B c = Bc Θ⋆4
(9)
n−1×n−1
T
where D = diag(c , I, I), I ∈ R is an identity matrix and Ac ∈ R3n−2×3n−2 , B c ∈ R3n−2 are the matrices of nonminimal realization of the reference model in the form X˙ m (t) = Am Xm (t) + B c r(t) (10a) ym (t) = CcT Xm (t) (10b) where r(t) is the reference signal and ym (t) is the reference model output. The reference model transfer function ym (s) ⋆ r(s) = Wm (s) has the relative degree nm = 2. The equation (8a) represents the controlled system. The term Bc Θ⋆c T DX(t) and the term Bc Θ⋆4 r(t) are added and subtracted to (8a), then ˙ X(t) = Ac X(t)+B c r(t)−Bc Θ⋆ T ω(t) +Bc u(t−τ ) (11) where Θ⋆ T = Θ⋆c T Θ⋆4 and ω T (t) = X T (t)DT r(t) has been introduced. Further the term Bc u(t) and the term B c ρ(t)uτ (t), where uτ (t) = u(t) − u(t − τ ) and ρ(t) ∈ R is a control algorithm parameter to be adapted, are added and subtracted to (11), which implies ˙ X(t) = Ac X(t) + B c r(t) + Bc u(t) − Θ⋆ T ω(t) (12) + B c ρ(t)uτ (t) − Θ⋆4 −1 uτ (t) − B c ρ(t)uτ (t) The proposed control law has the form u(t) = ΘT (t)ω(t) (13) 2n−1 where Θ(t) ∈ R is the vector of adapted parameters. Substituting (13) to (12) leads to ˙ X(t) = Ac X(t) + B c r(t) − B c ρ(t)uτ (t) ⋆T ⋆ −1 B (t) − Θ ρ(t) − Θ + B c ΘT ω(t) + uτ (t) c τ τ 4
(14)
Θ⋆4 −1 Θ(t)
Θ⋆τ
where Θτ (t) = and is defined analogically. Further, the adapted parameter errors are defined as ˜ τ (t) = Θτ (t) + Θ⋆ and ρ˜(t) = ρ(t) − Θ⋆ −1 . Moreover Θ τ 4 T ˜ τ (t) ρ˜(t) and ωτT (t) = ω T (t) uτ (t) are θτT (t) = Θ introduced. Then the equation (14) has the form ˙ X(t) = Ac X(t) + B c r(t) − B c ρ(t)uτ (t) + B c θτT (t)ωτ (t) (15) Since the controlled system relative degree equals two, the augmented reference model model is considered. This is well-known approach in the classical MRAC design. Equation (10a) is considered in the form X˙ m (t) = Am Xm (t) (16) T + B c r(t) + θτ (t) − L(s)θτ (t)L−1 (s) ωτ (t) where L(s) = (s + ς), ς ∈ R, ς > 0 is specified below.
Subtracting (16) from (15), the error equation is obtained in the form 454
e(t) ˙ = Ac e(t) + B c ρ(t)uτ (t) + B c θτT (t)ωτ (t) T − B c θτ (t) − L(s)θτ (t)L−1 (s) ωτ (t)
(17)
where e(t) = X(t)−Xm (t). Introducing the transformation ea (t) = e(t) + Xa (t) allows to split (17) into two equations e˙ a (t) = Ac ea (t) + B c θτT (t)ωτ (t) − B c θτ (t) − L(s)θτ (t)L−1 (s)
T
ωτ (t)
(18)
(19) X˙ a (t) = Ac Xa (t) + B c ρ(t)uτ (t) where a Smith-predictor-like filter has been introduced. The output equation of the filter has the form ya (t) = CcT Xa (t). Equation (18) can be rearranged to the form e˙ a (t) = Ac ea (t) + B c L(s)θτ T (t)L−1 (s) ωτ (t) (20a) ea1 (t) = CcT ea (t) (20b) where the output equation has been added. The system (20) is described by the transfer function in the form ea1 (s) = Wm (s) L(s)θτ T (s)L−1 (s) ωτ (s) (21) At this point a standard error equation suitable for a standard Lyapunov based output-feedback MRAC design has been obtained. Equation (21) can be written in the form ea1 (s) = Wm (s)L(s) θτ T (s)ωf (s) (22) where ωf (s) = L−1 (s)ωτ (s) and ς is chosen so that the Wm (s)L(s) is a Strictly Positive Real (SPR) transfer function. In view of (22) the control objective can be considered as follows: designing an adaptation law so that the augmented error ea1 (t) tends to zero as t → ∞ implies that the standard error e1 (t) = y(t) − ym (t) = CcT e(t) is bounded. Recall that the controlled system is assumed to be stable. This is referred as an adaptive model following (not a perfect model following). Using a well-understood adaptive control methods (Lyapunov based) it can be shown that the adaptation law in the form θ˙τ (t) = −Γea1 (t)ωf (t) (23) ensures that the control objective is satisfied and all of the closed-loop signals are bounded. In (23) Γ = diag (Γτ , γ), Γτ > 0 is an arbitrary diagonal matrix of appropriate ˙ τ (t) = dimension and γ > 0 is a scalar. It follows that Θ −Γτ ea1 (t)ωf f (t) and ρ(t) ˙ = −γea1 (t)uf f (t) (24) where uf f (s) = L−1 (s)uτ (s) and ωf f (s) = L−1 (s)ω(s). As defined above Θ(t) = Θ⋆4 Θτ (t), which implies ˙ Θ(t) = −sgn(Θ⋆4 )Γτ ea1 (t)ωf f (t) (25)
Therefore, a sign of Θ⋆4 has to be known, in other words the sign of controlled system high-frequency gain has to be known.
It can be shown that the signal ea1 (t) can be implemented in the form ea1 (t) = y(t) − ym (t) + ya (t) − [Wm (s)L(s)] sgn(Θ⋆4 ) L−1 (s) u(t) (26) −sgn(Θ⋆4 )ΘT (t)ωf f (t) + L−1 (s) ρ(t)uτ (t) −ρ(t)uf f (t))
11th IFAC ALCOSP July 3-5, 2013. Caen, France
3.2 Modification for Robust Adaptive Control The disturbance has not been considered in the previous section. However, since the disturbance is not rejected completely in the considered controlled system, the design of adaptive controller has to be modified to handle the disturbance influence. A switching σ-modification, which is well known approach in robust adaptive control, is considered. Therefore, the adaptation laws (24) and (25) are modified as follows. ρ(t) ˙ = σρ ρ(t) − γea1 (t)uf f (t) (27) ⋆ ⋆ ˙ Θ(t) = sgn(Θ4 )σΘ Θ(t) − sgn(Θ4 )Γτ ea1 (t)ωf f (t) (28)
where
σρ = and
0 if |ρ(t)| ≤ ρmax σρ0 otherwise
(29)
0 if |Θ(t)| ≤ Θmax (30) σΘ0 otherwise where ρmax , σρ0 , Θmax and σΘ0 are the design constants. σΘ =
3.3 A Heuristic Based Adaptive Disturbance Rejection Recall the definition ea (t) = e(t) + Xa (t). It follows that ea1 (t) = y(t) − ym (t) + ya (t), and finally ym (t) = y(t) + ya (t) − ea1 (t) (31) However, if the disturbance is present, i.e. Ψ⋆ T w(t) 6= 0 in (4), the output signal y(t) in (31) is impaired. Consequently, the signal ym (t) obtained using (31) is also impaired, which is caused only by the disturbance.
An ideal (non-impaired) case is represented by the reference model transfer function ⋆ ym (s) = Wm (s)r(s) (32) Therefore, a heuristic error can be defined in the form ⋆ ed (t) = ym (t) − ym (t) (33) It is clear, that if ed (t) = 0 the disturbance is not present or it is rejected. The adaptive disturbance rejection control law is proposed in the form ud (t) = ΨT (t)w(t) (34) 3 where Ψ(t) ∈ R are the parameters to be adapted. A control action ud (t) is subtracted from u(t), in other words, in (4) it can be written ∆v(t) = u(t) − ΨT (t)w(t). In view of (33) the adaptation law is proposed in the form ˙ Ψ(t) = σΨ Ψ(t) − γd ed (t)w(t) (35)
where γd ∈ R, γd > 0 and 0 if |Ψ(t)| ≤ Ψmax σΨ = (36) σΨ0 otherwise where Ψmax and σΨ0 are the design constants. The adaptation law (35) can be considered as a gradient based and modified via switching σ-modification. 4. MAIN RESULTS The proposed adaptive controller has been tuned and verified by means of the simulation experiments. The T1DM subject computer simulator has been used as the controlled system.
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The simulator is an own implementation of A Meal Glucose-Insulin Model by Chiara Dalla Man and coworkers, see Man et al. (2007); De Nicolao et al. (2011). As has been mentioned above the same simulator has been used for the identification of the controlled system model. Sampling period Ts = 15 min has been used in (1). Therefore the time-delays τ = 60 min, τ1 = 15 min and τ2 = 30 min are assumed to be known to the controller. The main objective in diabetes control is to keep the blood glucose concentration as tightly as possible within a normal glycemic range (3.8 – 10 mmol/l). The reference model used in the adaptive controller is chosen in the form 1.6 × 10−5 ym (s) = 2 r(s) (37) s + 0.008s + 1.6 × 10−5 i.e. two poles at −0.004 min−1 . As the reference signal r(t) a periodic square signal is used, with an amplitude 0.5 mmol/l (around the operating point GM b ) and with the period one day. Simulations are started in the steady state with the basal subcutaneous glucose concentration GM b . The tuning parameters of the adaptive controller such as the adaptation gains or the auxiliary filters are the same in all simulation experiments. The tuning parameters and the reference model have been chosen experimentally so that the same set can be used for each task considered in all simulations. A time period of three weeks is simulated. 4.1 Experiment No. 1 A control performance of the proposed controller with no disturbance present has been examined in the first simulation experiment. Therefore the heuristic based adaptive disturbance rejection has been turned off. Since the disturbance has not been present the switching σ-modification could be unused. In this experiment only the MRAC based part of controller has been used. The initial values of corresponding adapted parameters have been chosen experimentally and the same initial values have been used in all experiments. Results are shown in Fig. 2. As the figure clearly shows, the transient process of adaptation is over in three weeks. The model following (recall that the perfect model following has not been the control objective) has been achieved while the influence of the time-delay has been handled by controller. These results have verified the design of the MRAC based controller. However, a considered situation is not common from the diabetes control point of view. 4.2 Experiment No. 2 In the second experiment the disturbance has been considered in the form of meal ingestion. Each day the same meal protocol has been used as follows: a typical day life of the T1DM subject receiving a mixed meal, with 30 g of glucose over 15 minutes ingested at 8 a.m. (breakfast), 50 g of glucose over 15 minutes at noon (lunch) and 50 g of glucose over 15 minutes at 8 p.m. (dinner). This situation is more common in the diabetes control than the first experiment. However, the same controller setup as in the first experiment has been used, i.e. the heuristic based adaptive
11th IFAC ALCOSP July 3-5, 2013. Caen, France
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Fig. 4. Results of the simulation experiment No. 3 disturbance rejection has been turned off. Since the obvious presence of the disturbance, the use of switching σ-modification has been justified in this case. This has also guarantee the robust stability of the overall closedloop system. The control performance, however, has to be verified. The results are shown in Fig. 3. It can be seen, that the main control objective has not been satisfied, i.e. the glucose concentration in the normal glycemic range. Simulations for longer time interval than three weeks has shown, that the best performance, in the considered 456
situation, has been achieved in th 21st day and it has remained unchanged in next weeks. Moreover, during the adaptation the hypoglycemia has occurred in the first nine days, as emphasized in Fig. 3a. Nevertheless, it can be concluded that the performance after the tenth day is comparable to the diabetes control algorithms without a meal announcement considered (for example the classical PID controllers or the generalized predictive control algorithms reported in literature, see Miklovičová and Tárník (2012)).
11th IFAC ALCOSP July 3-5, 2013. Caen, France
4.3 Experiment No. 3 The same meal protocol as in the second experiment has been considered in the third experiment. The adaptive disturbance rejection, however, has been used in this experiment. Results in Fig. 4 have shown that after the adaptation transient, which has last only one day, the main control objective has been achieved. It can be concluded, that the heuristic adaptive disturbance rejection has a significant influence on the overall control performance. 5. CONCLUSION AND FUTURE WORK The presented adaptive controller design is based on the controlled system model inferred from the glucose kinetics minimal model, where in addition the subcutaneous tissues are modeled using the known input time-delay. The main results have shown that for the exact knowledge of this time-delay a satisfactory glucose control is obtained and the control objective is achieved. However, the influence of uncertainty in this time-delay has to be examined in future research. Our recent results have shown that for the stability the considered robust modification is adequate. Moreover, the control performance deterioration has shown to be acceptable from the diabetes control point of view. The same is true for the developed disturbance submodel, since there are also the time-delays, which are assumed to be known. However, in this case the uncertainty in the announced meal time has to be also considered in the future research. Nevertheless, the well-know importance of the disturbance rejection in the algorithms for diabetes control has been also shown in the main results. The benefits of the adaptive disturbance rejection in this case are obvious. REFERENCES De Nicolao, G., Magni, L., Dalla Man, C. and Cobelli, C. (2011), Modeling and control of diabetes: Towards the artificial pancreas, in ‘Proceedings of the 18th IFAC World Congress’. Hovorka, R., Shojaee-Moradie, F., Carroll, P. V., Chassin, L. J., Gowrie, I. J., Jackson, N. C., Tudor, R. S., Umpleby, A. M. and Jones, R. H. (2002), ‘Partitioning glucose distribution/transport, disposal, and endogenous production during ivgtt’, American Journal of Physiology - Endocrinology and Metabolism (282), E992–E1007. Magni, L., Raimondo, D., Man, C. D., Nicolao, G. D., Kovatchev, B. and Cobelli, C. (2009), ‘Model predictive control of glucose concentration in type i diabetic patients: An in silico trial’, Biomedical Signal Processing and Control 4(4), 338 – 346. Special Issue on Biomedical Systems, Signals and Control Extended Selected papers from the IFAC World Congress, Seoul, July 2008. Man, C., Raimondo, D., Rizza, R. and Cobelli, C. (2007), ‘Gim, simulation software of meal glucose–insulin model’, Journal of Diabetes Science and Technology 1(3). 457
Miklovičová, E. and Tárník, M. (2012), GPC for diabetes control without meal annoucement — control loop design and control performance study, in O. Arslan and S. Oprisan, eds, ‘Recent Advances in Mechanical Engineering and Automatic Control, Proceedings of the 3rd European Conference of Control (ECC ’12), Paris, France December 2-4, 2012’, WSEAS Press, pp. 58 – 63. Mirkin, B. and Gutman, P.-O. (2011), Adaptive following of perturbed plants with input and state delays, in ‘2011 9th IEEE International Conference on Control and Automation (ICCA)’, pp. 865 –870. Mirkin, B., Mirkin, E. L. and Gutman, P.-O. (2009), ‘State-feedback adaptive tracking of linear systems with input and state delays’, International Journal of Adaptive Control and Signal Processing 23(6), 567–580. Niculescu, S. and Annaswamy, A. M. (2003), ‘An adaptive smith-controller for time-delay systems with relative degree n⋆ ≤ 2’, Systems and Control Letters 49(5), 347 – 358. Ortega, R. and Lozano, R. (1988), ‘Globally stable adaptive controller for systems with delay’, International Journal of Control 47, 17–23. Tarnik, M. (2012), On adaptive posi-cast controller: Pmsm speed controller design, in ‘10th International Conference CONTROL OF POWER SYSTEMS 2012’. Yildiz, Y., Annaswamy, A., Kolmanovsky, I. V. and Yanakiev, D. (2010), ‘Adaptive posicast controller for time-delay systems with relative degree n∗ ≤ 2’, Automatica 46(2), 279 – 289.